This textbook provides an introduction to the mathematics on which modern cryptology is based. It covers not only public key cryptography, the glamorous component of modern cryptology, but also pays considerable attention to secret key cryptography, its workhorse in practice.
Modern cryptology has been described as the science of the integrity of information, covering all aspects like confidentiality, authenticity and non-repudiation and also including the protocols required for achieving these aims. In both theory and practice it requires notions and constructions from three major disciplines: computer science, electronic engineering and mathematics. Within mathematics, group theory, the theory of finite fields, and elementary number theory as well as some topics not normally covered in courses in algebra, such as the theory of Boolean functions and Shannon theory, are involved.
Although essentially self-contained, a degree of mathematical maturity on the part of the reader is assumed, corresponding to his or her background in computer science or engineering. Algebra for Cryptologists is a textbook for an introductory course in cryptography or an upper undergraduate course in algebra, or for self-study in preparation for postgraduate study in cryptology.
Author(s): Alko R. Meijer
Series: Springer Undergraduate Texts in Mathematics and Technology
Publisher: Springer
Year: 2016
Language: English
Pages: 311
1 Prerequisites and Notation
1.1 Sets
1.2 Products of Sets
1.3 Relations
1.4 Functions
1.5 Binary Operations
1.6 Cryptography
1.6.1 Encryption Mechanisms.
1.6.2 Confusion and Diffusion
1.6.3 Symmetric and Asymmetric Encryption
1.7 Notational Conventions
1.7.1 Floor and Ceiling
1.7.2 Fractional Part
1.7.3 Exclusive Or
1.7.4 Matrix Multiplication.
2 Basic Properties of the Integers
2.1 Divisibility
2.2 Ideals and Greatest Common Divisors
2.3 The Euclidean Algorithm
2.3.1 Stein’s gcd Algorithm
2.4 Congruences
2.5 Fermat’s Factoring Method
2.6 Solving Linear Congruences.
2.7 The Chinese Remainder Theorem
2.8 Some Number-Theoretic Functions
2.8.1 Multiplicative Functions
2.8.2 The Möbius Function.
2.8.3 Euler’s φ-Function
2.8.4 The Case n = p * q
3 Groups, Rings and Ideals
3.1 Groups
3.2 Subgroups
3.3 The Lattice of Subgroups
3.4 Cosets
3.5 Cyclic Groups
3.6 Fermat’s Little Theorem
3.7 Primality Testing
3.7.1 Miller’s Test
3.7.2 The Miller–Rabin Primality Test
3.8 Rings and Ideals
4 Applications to Public Key Cryptography
4.1 Public Key Encryption: The RSA Mechanism.
4.1.1 RSA
4.1.2 Breaking RSA
4.1.3 Using the Chinese Remainder Theorem
4.1.4 Public Key Infrastructures.
4.2 The Discrete Logarithm Problem (DLP)
4.2.1 Statement of the Problem
4.2.2 Exhaustive Search
4.2.3 Shanks’ Method
4.2.4 Pollard’s Rho Method
4.3 Diffie–Hellman Key Establishment
4.4 Other Applications of the DLP
4.4.1 ElGamal Encryption
4.4.2 The ElGamal Digital Signature Scheme
4.4.3 The Digital Signature Algorithm
4.4.4 A Zero Knowledge Proof
4.4.5 What Groups to Use?
5 Fields
5.1 Rings “Without” Ideals
5.2 Vector Spaces
5.3 Rings of Polynomials
5.3.1 Definitions
5.3.2 The Euclidean Algorithm Again
5.3.3 The “Remainder Theorem”
5.3.4 Irreducible Polynomials
5.4 Ring Homomorphisms
5.5 The Chinese Remainder Theorem Again
5.6 Construction of Finite Fields
6 Properties of Finite Fields
6.1 The Minimal Polynomial of an Element
6.2 More on the Structure of Finite Fields
6.3 Solving Quadratic Equations over GF.2n/ and the Trace Function.
6.4 Operations in Finite Fields
6.4.1 Addition in Zn
6.4.2 Inverses in Zn
6.4.3 Multiplication in GF(p^n)
6.4.4 Exponentiation
6.4.5 Inverses in GF(p^n)
6.5 Factoring Polynomials
6.5.1 Square-freeness
6.5.2 A Test for Irreducibility
6.5.3 Finding a Factor
6.6 The Discrete Logarithm Problem on a Finite Field
6.7 Elliptic Curves over a Finite field
7 Applications to Stream Ciphers
7.1 Introduction.
7.1.1 Stream Ciphers vs Block Ciphers
7.1.2 Design Principles
7.1.3 Terminology
7.2 Some Notes on Entropy
7.2.1 Entropy = Uncertainty
7.2.2 Conditional Entropy
7.2.3 Information
7.2.4 Redundancy
7.3 Perfect Secrecy
7.4 Linear Feedback Shift Registers
7.5 LFSRs: Further Theory
7.6 Galois Field Counters
7.6.1 Shift Registers in Feedforward Mode
7.6.2 Finite Field Multiplication
7.6.3 Galois Counters and Authenticated Encryption
7.7 Filter and Combining Functions
7.8 Linear Complexity
7.8.1 Introduction
7.8.2 The Berlekamp–Massey Algorithm
7.8.3 The Linear Complexity Profile
7.9 On the Design of Stream Ciphers
7.10 Combination Generators
7.11 Filter Generators.
7.12 Clock Controlled Generators
7.12.1 The Stop-and-Go Generator.
7.12.2 Alternating Step Generator.
7.12.3 Shrinking Generator
7.12.4 Self-Shrinking Generator.
7.12.5 Bitsearch Generators
8 Boolean Functions
8.1 Introduction.
8.2 The Algebraic Normal Form
8.3 The Walsh Transform
8.3.1 Hadamard Matrices
8.3.2 Definition of the Walsh Transform
8.3.3 Correlation with Linear Functions
8.3.4 Correlation Immunity
8.3.5 Linear Algebraic Gloss
8.4 Autocorrelation
8.5 Nonlinearity
8.6 Propagation Criteria
8.7 Linear Structures
8.7.1 Linearity.
8.7.2 Another Measure of Nonlinearity.
8.8 Bent Functions
8.8.1 The Simplest Bent function
8.8.2 The “Dot-Product” Bent Functions
8.8.3 The Maiorama Construction
8.8.4 Other Constructions
8.8.5 Extensions of Bent Functions
8.9 Algebraic Immunity
8.10 Completeness
8.11 The Discrete Fourier Transform
8.11.1 Introduction
8.11.2 Linear Complexity Revisited.
8.11.3 DFT over a Finite Field
9 Applications to Block Ciphers
9.1 Block Ciphers
9.2 Finite Fields in Rijndael
9.2.1 Linear Operations.
9.2.2 The Rijndael Substitution Box
9.2.3 Properties of the Rijndael S-box
9.2.4 Properties of the Rijndael Round Function.
9.3 Properties of the Function F(x) = x^(2^k+1)
9.3.1 Differential Uniformity.
9.3.2 Nonlinearity.
9.3.3 Degree
9.4 Perfect Nonlinear S-boxes
9.5 Diffusion
9.5.1 Introduction
9.5.2 Linear Codes
9.5.3 Error Detection and Correction
9.5.4 Some Properties of MDS Matrices
9.5.5 MDS Matrices in Block Ciphers
9.5.6 Examples of MDS Matrices.
9.5.7 Binary Codes
9.5.8 Boolean Functions and Codes
10 Number Theory in Public Key Cryptography
10.1 Secret Sharing: Shamir’s Mechanism
10.2 Access Structures and the Chinese Remainder Theorem
10.2.1 Access Structures
10.2.2 Disjunctive and Conjunctive Normal Forms
10.2.3 Yet Another Appearance of the CRT
10.2.4 The Conjunctive Normal Form
10.3 Quadratic Residues
10.4 The Legendre Symbol
10.5 The Quadratic Reciprocity Theorem
10.6 The Jacobi Symbol
10.7 Solving Quadratic Congruences
10.8 The Quadratic Residuosity Problem
10.9 Applications of Quadratic Residues
10.9.1 Rabin Encryption
10.9.2 Goldwasser–Micali Encryption
10.9.3 Coin Flipping by Telephone.
10.9.4 Oblivious Transfer
10.9.5 Exchanging Secrets with Oblivious Transfer.
10.9.6 Commitment
10.9.7 Blum–Blum–Shub Pseudo-Random Bit Generation
11 Where Do We Go from Here?
11.1 Further Reading
11.2 Further Topics
11.2.1 Elliptic Curves
11.2.2 Lattices
11.2.3 Homomorphic Cryptography
11.2.4 Post-Quantum Cryptography.
11.3 Even More Further Reading
A Probability.
A.1 Definitions
A.1.1 Introduction
A.1.2 Conditional Probability.
A.1.3 Independence of Events
A.1.4 Mode, Mean and Variance
A.2 Discrete Probability Distributions
A.2.1 The Uniform Distribution
A.2.2 The Binomial Distribution
A.2.3 The Bernoulli Distribution
A.2.4 The Poisson Distribution
A.2.5 The Geometric Distribution
A.3 Continuous Probability Distributions
A.3.1 The Uniform Distribution
A.3.2 The Normal Distribution
A.3.3 The Exponential Distribution
A.4 The Central Limit Theorem
A.5 The Chi-Squared Distribution
A.6 The Birthday “Paradox”
A.7 Bayes’ Theorem
Index
